Various and Sundry

The Institute in Princeton has its usual summer program designed to train graduate students and postdocs in string theory. The schedule and lecture notes are here.

On the opposite coast, with an opposite point of view about particle physics, there’s the SLAC Summer Institute, which is on LHC physics. The program and lecture notes are here. One of the organizers, JoAnne Hewett, has a posting about this at Cosmic Variance.

Last weekend there was a conference entitled Under the Spell of Physics, in honor of ‘t Hooft’s 60th birthday. Many of the talks sound interesting; here’s the program, but unfortunately the talks are not online. From what I hear ‘t Hooft remains quite skeptical about string theory, Polyakov said that current ideas about how to apply string theory to nature are wrong, and the lack of progress in fundamental theory was a concern of many of the participants.

I’ve been thinking a lot about BRST recently, and happened to run across the Wikipedia entry for BRST Formalism. The entry had something I hadn’t seen before, a banner announcing that “This article or section may be confusing or unclear for some readers, and should be edited to rectify this”, and that the attention of an expert and a complete rewrite was needed. I have to say that I feel that way about most of the literature on BRST…

The Cao-Zhu paper with a proof of Poincare/Geometrization is now out in paper copies of the Asian Journal of Mathematics, but still is not on the journal’s web-site. I hear that someone who called them to ask about this was told that they’re trying to make some money by selling the paper copies of this particular issue. Many libraries are now only paying for on-line access to journals like this, not sure what happens in this case. Today’s Wall Street Journal had an article by Sharon Begley about the Poincare proof story.

FQXI was supposed to announce the winners of its Templeton-funded grants this past weekend, but still nothing on their web-site. It will be interesting to see what their choices are for fundamental research in physics that is not being supported by the usual channels.

Update: The FQXI web-site now says they’ll be publicly announcing grants on Monday, July 31.

THE ASIAN JOURNAL IS BEING VERY VERY EVIL AND GREADY, WE SHOULD BE ABLE TO SEE POINCARE PROOF ONLINE. ALSO IT DOES NOT HELP TO ADRESS THE CYNICISM ABOUT THE PROOF. ARE THERE ANY FAMOUS PEOPLE OF GEOMETRY (NOT YAU WHO SEEMS TO BE DOING POLITICS) WHO HAVE SCRUTINIED THIS PROOF?

Are there other good introductions out there on the arxiv or on the net? I think BRST is of foundational importance. And it would be nice to further develop a geometric understanding, especially in its application for nonabelian Yang-Mills fields.

There are some hints in the earlier literature of a geometry behind BRST, with ghost fields literally added to connections in some sort of hybrid bundle, but my understanding is sketchy.

The neat thing that’s hinted at, at least for BRST applied to nonabelian gauge theory, is that the BRST operator, s, is a sort of exterior derivative (added to the exterior derivative on the base manifold) that acts in a discrete “direction”, and the ghosts are the components of the connection in that direction.

I believe you can get a good intuitive understanding of BRST (although certainly not in its full generality) from studying Feynman diagrams. I think ‘t Hoof did some of this in his very early work on renormalization of gauge theories.

Like a lot of people, I appreciated the original BRS work
since it can be used to generate a useful form of gauge-theory
Ward identities to prove renormalizability. Beyond that, BRST just seems a way of eliminating unphysical degrees of freedom.

The fascination with BRST by many people seems a little questionable to me, however. Some people actually think it is fundamental, but I can’t see how this could be true. It is basically a perturbative supersymmetry. It doesn’t work with a nonperturbative regularization – which means the lattice gauge theory (there is no other successful non-perturbative regularization). Herbert Neuberger wrote an interesting paper on this, some years ago.

BRST may be interesting mathematics, but its utility in much of physics is limited to to perturbation theory. This makes it an important tool, but not a path to a fundamental understanding of non-Abelian gauge theories. I am aware that it has a lot of utility in other areas, but I can’t help suspecting that it can’t be used beyond perturbation theory even in those cases.

It certainly is true and interesting that you don’t seem to need to gauge-fix in the lattice theory, but I still think there should be a non-perturbative version of BRST, and understanding it might teach one something interesting.

One reason for believing this is that one aspect of (quantum) BRST is something well-known to mathematicians: Lie algebra cohomology. The Lie algebra cohomology of a representation is just about by definition the invariant part of the representation. So, this is a mathematical gadget precisely designed to pick out the physical subspace in the Hilbert space of a gauge theory, and the existence of such a gadget should be a very general feature of the quantum theory, not dependent on perturbation theory. Whether one can construct it explicitly non-perturbatively is another question.

In the lattice theory, one picks out the gauge invariant subspace by integrating over the gauge group, and one can get away with this because for a finite lattice this is compact.

I’m not sure which Neuberger paper you’re referring to. I’d be interested to hear more from anyone who knows of good references to papers where people try to do BRST on the lattice.

I think the source of our disagreement is technical, not philosophical.
Ultimately what the lattice tells you is that the Lie group is more important than the lie Algebra. I think that there is something fundamentally wrong with trying to use Lie algebras to define gauge theory nonperturbative.

I can’t prove my assertion, but I have several of arguments why it
is probably true. For example, strong-coupling calculations are the key to why QCD confines, at least for large bare coupling. Such expansions make no sense without a compact field degree of freedom.

Neuberger’s paper is in the hep-lat archive. I think he actually
wrote several papers. In the first one he was trying to prove BRST, but
actually proved a no-go theorem in the second. My memory on
this is vague, unfortunately.

One possible way out – generalize BRST to compact spaces. If you
know some compact version of supersymmetry, perhaps it can be
dome.

You may be right, the difference between the Lie algebra and the group may be crucial. But even so, you should in principle be able to define the part of the Hilbert space invariant just under infinitiesimal gauge transformations. The part invariant under the gauge group should be a subspace of this. Understanding the difference between the two spaces would be interesting.

I think of BRST as Lie algebra cohomology, there are various versions of cohomology that use instead the full group. But in the compact case, they’re really the same. Hmm, actually in the lattice case, since the group is compact, presumably one can show that, for a connected group, being invariant under the Lie algebra is the same as being invariant under the group. Have to think more about that…

Well, that in a way is the point. Gauge transformations on the lattice
sit in a Lie group SU(N) to the power of the number of lattice sites.
This is crucially different from the Lie algebra. It is not just the fields
which are compact.

By the way, this is a bit of self-promotion (parties disinterested in the confinement problem should ignore it), but I have recently put out a paper on confinement in lattice gauge theory at weak coupling. The compactness of the gauge group is crucial there too, just as at strong coupling. The gauge theory is in 2+1 dimensions, the coupling are anisotropic, it’s only SU(2), but it’s the only case I know where things seems to work for unbroken non-Abelian gauge groups. None of
this will work unless the theory is on a lattice (and I think it does
work, if the theory is on a lattice).

OK, sorry about the advertisement, but I still think you need a BRST which is some sense is compact, otherwise important physics is missed.

>The Cao-Zhu paper with a proof of Poincare/Geometrization is >now out in paper copies of the Asian Journal of Mathematics, but >still is not on the journal’s web-site. I hear that someone who >called them to ask about this was told that they’re trying to make >some money by selling the paper copies of this particular issue. >Many libraries are now only paying for on-line access to journals >like this, not sure what happens in this case.

This sounds really strange.

The online access to the papers in AJM isn’t free. So does this actually mean that people are not going to get what they are paying for?

By the way, there is a non-lattice way to define gauge theories
nonperturbatively in 2+1 dimensions, due to Karabali and Nair. They use a parametrization of the gauge orbits which is a kind of gauge fixing (someone showed this gauge fixing is closely related to background-field gauge). Strong coupling-expansions, done
by diagonalizing the kinetic term of the Hamiltonian are possible. There is a severe Gribov problem in generalizing this idea to 3+1, BUT…

Is there a BRST for Karabali and Nair’s parametrization? I doubt it.
It seems worth looking into though…

I don’t know what AJM is doing about this. Maybe they’re sending paper copies to people who just paid for on-line access, maybe the paper will appear online, just later. My information that they are doing this to raise money is third hand, and not necessarily completely reliable. However I do have first hand information that the paper version exists (it’s downstairs in our library), and that the on-line one doesn’t., and that’s kind of unusual.

1. When you say that BRST is perturbative, you mean it is only useful in perturbation theory, or that it is only defined perturbatively? The latter seems surprising to me. Maybe this is just then statement that on the lattice there is no need for gauge fixing?

2. Karabali and Nair work in the temporal gauge, and then use a clever parametrization of the gauge-fixed degrees of freedom, naively I would expect BRST to be more or less unaffected, even if the clever parametrization allows for some strong coupling results. On the other hand I am not sure what you mean in the statement that their parametrization is similar to background field gauge…

The basic idea behind BRST applies more generally than to gauge symmetries. Whenever one has a complicated factor space, one can try to replace it by a sequence of simple spaces, such that the original space is recovered as the cohomology space H^0.

In BRST, one constructs the space of connections modulo gauge transformations in this way. In BV, one instead constructs phase space (even without gauge symmetries), as the space of histories modulo Euler-Lagrange equations. Just as one may coordinatize the space of connections module gauge transformations by fixing a gauge, one may coordinatize phase space by fixing a foliation – a history q(t) is specified by q and p = dq/dt at t = 0. But BRST teaches us that one obtains a cleaner and simpler description by cohomological means, where one does not have to make such arbitrary choices. Physics is hard as it is, without introducing spurious complications by hand.

One such complication, which has been discussed at length in connection with LQG, is the difference between the 3-diffeo and Hamiltonian constraints, which is an artefact of the introduction of a foliation. In cohomological and thus covariant formulations, both are part of the 4-diffeo constraint, because no foliation needs to be introduced.

Formally BRST is fine in unregularized Yang-Mills. In perturbation
theory, it is still fine. If you try to define the theory beyond perturbation theory, Neuberger’s theorem says (at least on the lattice) there are problems. My prejudice (and I stress that it is
only that) is that this makes BRST useless in understanding problems like quark confinement.

Karabali and Nair do more than a temporal gauge fixing. They
also solve Gauss’s law. The resulting variables describe gauge
orbits. In this sense, they have a physical gauge condition, like
axial or Coulomb gauge, but with the Gribov problem eliminated. There are other ways to do similar things, but I was expressing doubt (not a proof!) that BRST is satisfied after this further gauge fixing.

My comment about background field gauge is a little technical.
It is the gauge in which a variation of the gauge connection
has vanishing covariant divergence. This was first discussed
by Babelon and Viallet around 1980. It is a useful gauge for
discussing orbit space geometry (inner products, curvature,….).

Peter said
The Lie algebra cohomology of a representation is just about by definition the invariant part of the representation. So, this is a mathematical gadget precisely designed to pick out the physical subspace in the Hilbert space of a gauge theory, and the existence of such a gadget should be a very general feature of the quantum theory, not dependent on perturbation theory.

This reminds me of the role of Lie derivatives in GR. Although unnecessary to formulate the theory, the Lie derivatives of this and that represent actual physical things, e.g. transport phenomena. I wonder if this is coincidental.

I must agree with Qwerty’s assessment: “YAKOB SINAIS ARTICLE IS DISAPPOINTING”. Mathematicians and Physicists are not like cats and dogs, they are more like husband and wife (I’m not sure which is which), they often argue, but when they work together, the result can bring a new life into being.

What is curious is that way back in the good old days, people like Newton were considered mathematicians (and natural philosophers). Newton worked on optics ~ designed the Newtonian reflector (and dabbled in alchemy). Gauss was a mathematician, yet the unit of magnetic field is named after him. Many famous mathematicians (Bernoullis, Lagrange, Laplace, etc) worked on problems of physics — witness the classic pdes = Laplace, heat, wave, all motivated by physics. The dichotomy (if that is the right word) between physicists and mathematicians seems to be a modern (20th century?) phenomenon.

If my own marriage is anything to go by, physicists are husbands. They are willing to be pragmatic about things. The wife will insist on every nitpicking detail.

The rumors of AJM trying to make money by selling copies of the current issue seem to be corroborated by the bottom of this webpage, wherein we learn that we can have the issue for the low low price of $69.

I’m not sure what exactly online subscribers to AJM (or the other International Press journals for that matter) are paying for, since as near as I can tell all the articles seem to be freely downloadable by anyone.

“I’m not sure what exactly online subscribers to AJM (or the other International Press journals for that matter) are paying for, since as near as I can tell all the articles seem to be freely downloadable by anyone.”

Since it is freely available online, that must reduce the number of printed subscription contributors to a few libraries. Hence the unit cost of printing is very high. It’s down to the economy of scale: offset litho only becomes cheap when the total sales dwarf the high cost of film-setting (producing printing plates).

>I’m not sure what exactly online subscribers to AJM (or the other International Press journals for that matter) are paying for, since as near as I can tell all the articles seem to be freely downloadable by anyone.

Well yeah, that was kind of my point…they do have an online subscription option, but I can’t see why anyone would choose to buy one when nonsubscribers can download all the papers for free (granted, now with the exception of the Cao-Zhu paper).

Unless of course the idea is that when International Press decides for whatever reason to deny open access to some article (like Cao-Zhu) the online subscribers would still get it–but this seems like it would be too rare and unforeseeable to cause anyone to subscribe.